CONSISTENCY OF AIC AND BIC IN ESTIMATING THE NUMBER OF SIGNIFICANT COMPONENTS IN HIGH-DIMENSIONAL PRINCIPAL COMPONENT ANALYSIS

In this paper, we study the problem of estimating the number of significant components in principal component analysis (PCA), which corresponds to the number of dominant eigenvalues of the covariance matrix of p variables. Our purpose is to examine the consistency of the estimation criteria AIC and...

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Bibliographic Details
Published in:The Annals of statistics 2018-06, Vol.46 (3), p.1050-1076
Main Authors: Bai, Zhidong, Choi, Kwok Pui, Fujikoshi, Yasunori
Format: Article
Language:English
Subjects:
Asymptotic methods
Computer simulation
Consistency
Covariance matrix
Criteria
Dimensional analysis
Eigenvalues
Estimating techniques
Estimation
Information theory
Matrix theory
Normality
Population distribution
Principal components analysis
Studies
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Summary:In this paper, we study the problem of estimating the number of significant components in principal component analysis (PCA), which corresponds to the number of dominant eigenvalues of the covariance matrix of p variables. Our purpose is to examine the consistency of the estimation criteria AIC and BIC based on the model selection criteria by Akaike [In 2nd International Symposium on Information Theory (1973) 267–281, Akadémia Kiado] and Schwarz [Estimating the dimension of a model 6 (1978) 461–464] under a high-dimensional asymptotic framework. Using random matrix theory techniques, we derive sufficient conditions for the criterion to be strongly consistent for the case when the dominant population eigenvalues are bounded, and when the dominant eigenvalues tend to infinity. Moreover, the asymptotic results are obtained without normality assumption on the population distribution. Simulation studies are also conducted, and results show that the sufficient conditions in our theorems are essential.
ISSN:0090-5364
2168-8966
DOI:10.1214/17-AOS1577